You can now access the newly updated Weather information from Google without having to first open the Google or News & Weather apps. However, you will have to use Google to access the weather the old-fashioned way one more time, but you'll have the option to place a shortcut on your home screen to access the weather directly going forth.
Open up the Google app and either say "OK Google" or press the microphone icon to activate voice search. Then, say something along the lines of "Show me the weather" or "What's the weather like in [specific area]," etc. You can also just type "weather" into the search bar, but c'mon, that's kind of boring.
You should see the weather pop up first on the results page.
Scroll down to the bottom, and you'll see an arrow icon and some text below it that says "Humidity, wind, sunrise, and more details." Tap the arrow icon to expand the info, and you'll be greeted by a little frog dude.
The "Add this location to your list?" option lets you add multiple locations to the Weather app for easy reference. If you do add some, you can access them by tapping on the search box.
Now you just have to tap on the hamburger (three lines) menu at the top-left of the screen. Under the "Weather" section, you'll see the option to switch between Fahrenheit and Celsius, and under that, an option to add Weather to your home screen.
Just tap on that, and now you'll have a shortcut to Google's Weather app on your home screen. From there, you can put the shortcut wherever you want.
Unfortunately, the feature is only in a testing phase now, so not everybody is seeing it. There's also a possibility that the weather shortcut is only available to phones running Marshmallow 6.0 or higher. While it's certainly possible that Google will bring the feature to older versions of Android, we'll just have to wait and see at this point.